Problem 87-11: Monotonicity of Bessel functions
Citation for published version (APA):
Rienstra, S. W. (1987). Problem 87-11: Monotonicity of Bessel functions. SIAM Review, 29(3), 469-.
https://doi.org/10.1137/1029078
DOI:
10.1137/1029078
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Published: 01/01/1987
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SIAM REVIEW
Vol.29, No. 3,September1987
(C)1987 Society for Industrial and Applied Mathematics
OO4
PROBLEMS AND SOLUTIONS
EDITED
BY
MURRAY S. KLAMKIN
COLLABORATING EDITORS: CECIL
C. ROUSSEAU OTTO G. RUEHR
All
problems and solutions
should be
sent,typewrittenin duplicate,to
Muray S.
Klamkin,
Department
of
Mathematics,
University
of
Alberta, Edmonton, Alberta, Canada
T6G 2G
1.
An
asterisk placed beside
a
problem number indicates that the problem
was
submitted without solution.
Proposers
and solvers whose
solutions
are
published will
receive
5
reprints
of
the
corresponding problem
section.
Other
solvers
will
receive
just
one
reprint
provided a
self-addressed
stamped
(U.S.A.
or
Canada)
envelope
is
enclosed. Proposers
and
solvers
desiring acknowledgment
of
their contributions
should include
a
self-addressed
stamped postcard
(no
stamp
necessary outside the
U.S.A.
and
Canada).
Solutions should
be received by December 3
I,
1987.
PROBLEMS
Monotonicity of Bessel Functions
Problem 87-11", by S.
W.
RIENSTRA (Katholieke Universiteit,
Nijmegen, the
Nether-lands).
The
eigenvalue equation
related
to
a problem
of
sound propagation in
hard-walled annular ducts is given by
x2lJf,(x)Y[,(xh)
with
n---0, 1,
2,
...,
0
<
h
<
1,
where
Jn’
and
Y
denote
derivatives
of
Bessel
func-tions,
while
h
is
the ratio between the inner and outer duct
radii,
we
are
interested in
the solutions
x
a(h)
as
a
function
of
h.
In
particular, we want to show that
d.
f()
(x)
+
r’’
(x)
-=hlf(ah)-f(a)}
where
f(x)=
n2/x
2
is always finite or
f(ah)
f(a)
0.
If
n
=0,
the latter is
true
since
3(x)
Jl(X)
2
+
Yl(X)
2
is a
decreasing function
].
It
is
also true
for
n ->
if
ah
<
n and
c
>
n
(ct
_-< n does not
occur).
If
ah
n,
dc/dh
O.
Finally, it will also be
true
for
the
case
cth
>
n if
f(x)
is decreasing
for x
>
n.
In
view
of
numerical evidence
that this is so
for
n
0,
1,
...,
100,
it is conjectured
to
be
true.
Prove
or
disprove.
REFERENCE
G. N.
WATSON, A
Treatise on
the Theory
of
Bessel
Functions,
Cambridge University
Press, New
York,
1948, p. 446.
A
Column
Vector
Problem
From
Numerical Integration
Problem 87-12", by
M. M.
CHAWLA (Indian
Institute
of Technology,
New
Delhi,
India).
Let
w(x)
>
0 be
a
weight function defined
on
[a,
b]
with associated
orthonormal
polynomials
p*(x),
n >-_
O.
Let {Xk
}ffo
denote the
N
+
Gaussian abscissae
(the
zeros
of
pv+(x))
and
let
{Xk}v_-o
be the corresponding
(Gaussian)
weights
for
the
(N+
1)-point
(Gaussian)
quadrature
formula for
fba
W(x)f(x)
dx.
469